Let $\omega$ denote the set of non-negative integers and let $[\omega]^\omega$ be the collection of infinite subsets of $\omega$.

If $S\in [\omega]^\omega$ and $A\subseteq \omega$ we say that $A$ is *well-balanced with respect to* $S$ if $$\lim_{n\to\infty}\frac{|A\cap S\cap \{1,\ldots,n\}|}{|S\cap\{1,\ldots,n\}|+1} = 1/2.$$ Intuitively speaking, this means that $A$ contains "half of the members of $S$" (which also implies that $A$ is infinite.)

**Question.** Given ${\frak S}\subseteq [\omega]^\omega$ with $|{\frak S}| = \aleph_0$, is there $A\in[\omega]^\omega$ such that $A$ is well-balanced with respect to every member of ${\frak S}$?